Currently the best constraints on primordial non-Gaussianity come from the CMB bispectrum. In the near future, several CMB experiments will improve on these constraints, predominantlyIn addition the sensitivity to polarization will improve which will double the number of modes and, as our analysis shows, when combined with temperature measurements can result in non-trivial improvements on shapes with \(\Delta _c > 1/2\) . by increasing the spatial resolution, reaching to higher \(\ell _{\rm max}\)  [1]}, [2]}, [3]}. An often quoted threshold of any type of non-Gaussianity is to reach \(\sigma _{f_{\rm NL}} \sim 1\)  [4]}. Given the current bounds on orthogonal and equilateral non-Gaussianity \(\sigma _{f_{\rm NL}} \sim \mathcal {O}(50)\) , reaching that threshold will be challenging if not downright impossibleAn estimate suggests that this would require \(\ell _{\rm max} > 25000\)  for orthogonal and \(\ell _{\rm max} > 100000\)  for equilateral, requiring a CMB dish of \(\mathcal {O}(100)\)  meters in diameter with a focal plane loaded with detectors. This assumes we would have unconstrained access to primary modes, which we know is not the case already for \(\ell _{\rm max}\sim \)  few thousand. We should note that this extrapolation relies on the orthogonal template of Eq. (REF ). The correct orthogonal shape should have \(\Delta = 2\)  (see App. B of Ref. [5]}) and the expected scaling would be similar to the equilateral shape. with CMB measurements alone. While this is a somewhat pessimistic reading of our analysis, the fact that the bispectrum is not the only measure of non-Gaussianity, and that spectra with \(N>3\)  can exhibit very favorable scaling with resolution, implies that the CMB can certainly contribute to the search for primordial non-Gaussianity in general. For example, graviton exchange trispectra are qualitatively similar to \(\tau _{\rm NL}\) -like non-Gaussianity, while the corresponding bispectra are equilateral-like. Of course, this example poses a challenging detection regardless, due to the overall coefficient being Planck suppressed. It would be interesting to consider other examples displaying a similar mismatch of scaling behavior of the collapsed trispectrum vs. squeezed bispectrum.
